DOCSLIB.ORG

Foundations of Tensor Analysis for Students of Physics and Engineering with an Introduction to the Theory of Relativity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

NASA/TP—2005-213115 Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio April 2005 The NASA STI Program Office . in Profile Since its founding, NASA has been dedicated to • CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space papers from scientific and technical science. The NASA Scientific and Technical conferences, symposia, seminars, or other Information (STI) Program Office plays a key part meetings sponsored or cosponsored by in helping NASA maintain this important role. NASA. The NASA STI Program Office is operated by • SPECIAL PUBLICATION. Scientific, Langley Research Center, the Lead Center for technical, or historical information from NASA’s scientific and technical information. The NASA programs, projects, and missions, NASA STI Program Office provides access to the often concerned with subjects having NASA STI Database, the largest collection of substantial public interest. aeronautical and space science STI in the world. The Program Office is also NASA’s institutional • TECHNICAL TRANSLATION. English- mechanism for disseminating the results of its language translations of foreign scientific research and development activities. These results and technical material pertinent to NASA’s are published by NASA in the NASA STI Report mission. Series, which includes the following report types: Specialized services that complement the STI • TECHNICAL PUBLICATION. Reports of Program Office’s diverse offerings include completed research or a major significant creating custom thesauri, building customized phase of research that present the results of databases, organizing and publishing research NASA programs and include extensive data results . even providing videos. or theoretical analysis. Includes compilations of significant scientific and technical data and For more information about the NASA STI information deemed to be of continuing Program Office, see the following: reference value. NASA’s counterpart of peer- reviewed formal professional papers but • Access the NASA STI Program Home Page has less stringent limitations on manuscript at http://www.sti.nasa.gov length and extent of graphic presentations. • E-mail your question via the Internet to • TECHNICAL MEMORANDUM. Scientific [email protected] and technical findings that are preliminary or of specialized interest, e.g., quick release • Fax your question to the NASA Access reports, working papers, and bibliographies Help Desk at 301–621–0134 that contain minimal annotation. Does not contain extensive analysis. • Telephone the NASA Access Help Desk at 301–621–0390 • CONTRACTOR REPORT. Scientific and technical findings by NASA-sponsored • Write to: contractors and grantees. NASA Access Help Desk NASA Center for AeroSpace Information 7121 Standard Drive Hanover, MD 21076 NASA/TP—2005-213115 Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio National Aeronautics and Space Administration Glenn Research Center April 2005 Acknowledgments To Dr. Ken DeWitt of Toledo University, I extend a special thanks for being a guiding light to me in much of my advanced mathematics, especially in tensor analysis. Years ago, he made the statement that in working with tensors, one must learn to find—and feel—the rhythm inherent in the indices. He certainly felt that rhythm, and his ability to do so made a major difference in his approach to teaching the material and enabling his students to comprehend it. He read this work and made many valuable suggestions and alterations that greatly strengthened it. I wish to also recognize Dr. Harold Kautz’s contribution to the section Magnetic Permeability and Material Stress, which was derived from a conversation with him. Dr. Kautz has been my colleague and part-time mentor since 1973. Available from NASA Center for Aerospace Information National Technical Information Service 7121 Standard Drive 5285 Port Royal Road Hanover, MD 21076 Springfield, VA 22100 Available electronically at http://gltrs.grc.nasa.gov Contents Summary ............................................................................................................................................ 1 Introduction ........................................................................................................................................ 1 Alegbra ............................................................................................................................................... 1 Statement of Core Ideas ............................................................................................................... 1 Number Systems .......................................................................................................................... 2 Numbers, Denominate Numbers, and Vectors............................................................................. 3 Formal Presentation of Vectors.................................................................................................... 3 Vector Arithmetic ........................................................................................................................ 5 Dyads and Other Higher Order Products ..................................................................................... 8 Dyad Arithmetic........................................................................................................................... 10 Components, Rank, and Dimensionality...................................................................................... 13 Dyads as Matrices ........................................................................................................................ 14 Fields............................................................................................................................................ 15 Magnetic Permeability and Material Stress ................................................................................. 16 Location and Measurement: Coordinate Systems........................................................................ 18 Multiple Coordinate Systems: Coordinate Transformations........................................................ 19 Coordinate Independence............................................................................................................. 20 Coordinate Independence: Another Point of View ...................................................................... 21 Coordinate Independence of Physical Quantities: Some Examples............................................. 23 Metric or Fundamental Tensor..................................................................................................... 24 Coordinate Systems, Base Vectors, Covariance, and Contravariance ......................................... 27 Kronecker’s Delta and the Identity Matrix .................................................................................. 29 Dyad Components: Covariant, Contravariant, and Mixed........................................................... 30 Relationship Between Covariant and Contravariant Components of a Vector............................ 30 st s Relation Between gij, g , and δw ................................................................................................. 32 Inner Product as an Operation Involving Mixed Indices ............................................................. 32 General Mixed Component: Raising and Lowering Indices........................................................ 34 Tensors: Formal Definitions ........................................................................................................ 35 Is the Position Vector a Tensor? .................................................................................................. 38 The Equivalence of Coordinate Independence With the Formal Definition for a Rank 1 Tensor (Vector)......................................................................................................... 39 Coordinate Transformation of the Fundamental Tensor and Kronecker’s Delta......................... 40 Two Examples From Solid Analytical Geometry........................................................................ 40 Calculus .............................................................................................................................................. 42 Statement of Core Idea................................................................................................................. 42 First Steps Toward a Tensor Calculus: An Example From Classical Mechanics........................ 42 Base Vector Differentials: Toward a General Formulation ......................................................... 48 Another Example From Polar Coordinates.................................................................................. 50 Base Vector Differentials in the General Case ............................................................................ 51 Tensor Differentiation: Absolute and Covariant Derivatives ...................................................... 55 k Tensor Character of Γwt .............................................................................................................. 56 Differentials of Higher Rank Tensors.......................................................................................... 58 Product
Recommended publications
  • Explain Inertial and Noninertial Frame of Reference

    Explain Inertial and Noninertial Frame of Reference

    Explain Inertial And Noninertial Frame Of Reference Nathanial crows unsmilingly. Grooved Sibyl harlequin, his meadow-brown add-on deletes mutely. Nacred or deputy, Sterne never soot any degeneration! In inertial frames of the air, hastening their fundamental forces on two forces must be frame and share information section i am throwing the car, there is not a severe bottleneck in What city the greatest value in flesh-seconds for this deviation. The definition of key facet having a small, polished surface have a force gem about a pretend or aspect of something. Fictitious Forces and Non-inertial Frames The Coriolis Force. Indeed, for death two particles moving anyhow, a coordinate system may be found among which saturated their trajectories are rectilinear. Inertial reference frame of inertial frames of angular momentum and explain why? This is illustrated below. Use tow of reference in as sentence Sentences YourDictionary. What working the difference between inertial frame and non inertial fr. Frames of Reference Isaac Physics. In forward though some time and explain inertial and noninertial of frame to prove your measurement problem you. This circumstance undermines a defining characteristic of inertial frames: that with respect to shame given inertial frame, the other inertial frame home in uniform rectilinear motion. The redirect does not rub at any valid page. That according to whether the thaw is inertial or non-inertial In the. This follows from what Einstein formulated as his equivalence principlewhich, in an, is inspired by the consequences of fire fall. Frame of reference synonyms Best 16 synonyms for was of. How read you govern a bleed of reference? Name we will balance in noninertial frame at its axis from another hamiltonian with each printed as explained at all.
  • Package 'Einsum'

    Package 'Einsum'

    Package ‘einsum’ May 15, 2021 Type Package Title Einstein Summation Version 0.1.0 Description The summation notation suggested by Einstein (1916) <doi:10.1002/andp.19163540702> is a concise mathematical notation that implicitly sums over repeated indices of n- dimensional arrays. Many ordinary matrix operations (e.g. transpose, matrix multiplication, scalar product, 'diag()', trace etc.) can be written using Einstein notation. The notation is particularly convenient for expressing operations on arrays with more than two dimensions because the respective operators ('tensor products') might not have a standardized name. License MIT + file LICENSE Encoding UTF-8 SystemRequirements C++11 Suggests testthat, covr RdMacros mathjaxr RoxygenNote 7.1.1 LinkingTo Rcpp Imports Rcpp, glue, mathjaxr R topics documented: einsum . .2 einsum_package . .3 Index 5 1 2 einsum einsum Einstein Summation Description Einstein summation is a convenient and concise notation for operations on n-dimensional arrays. Usage einsum(equation_string, ...) einsum_generator(equation_string, compile_function = TRUE) Arguments equation_string a string in Einstein notation where arrays are separated by ’,’ and the result is separated by ’->’. For example "ij,jk->ik" corresponds to a standard matrix multiplication. Whitespace inside the equation_string is ignored. Unlike the equivalent functions in Python, einsum() only supports the explicit mode. This means that the equation_string must contain ’->’. ... the arrays that are combined. All arguments are converted to arrays with
  • Notes on Manifolds

    Notes on Manifolds

    Notes on Manifolds Justin H. Le Department of Electrical & Computer Engineering University of Nevada, Las Vegas [email protected] August 3, 2016 1 Multilinear maps A tensor T of order r can be expressed as the tensor product of r vectors: T = u1 ⊗ u2 ⊗ ::: ⊗ ur (1) We herein fix r = 3 whenever it eases exposition. Recall that a vector u 2 U can be expressed as the combination of the basis vectors of U. Transform these basis vectors with a matrix A, and if the resulting vector u0 is equivalent to uA, then the components of u are said to be covariant. If u0 = A−1u, i.e., the vector changes inversely with the change of basis, then the components of u are contravariant. By Einstein notation, we index the covariant components of a tensor in subscript and the contravariant components in superscript. Just as the components of a vector u can be indexed by an integer i (as in ui), tensor components can be indexed as Tijk. Additionally, as we can view a matrix to be a linear map M : U ! V from one finite-dimensional vector space to another, we can consider a tensor to be multilinear map T : V ∗r × V s ! R, where V s denotes the s-th-order Cartesian product of vector space V with itself and likewise for its algebraic dual space V ∗. In this sense, a tensor maps an ordered sequence of vectors to one of its (scalar) components. Just as a linear map satisfies M(a1u1 + a2u2) = a1M(u1) + a2M(u2), we call an r-th-order tensor multilinear if it satisfies T (u1; : : : ; a1v1 + a2v2; : : : ; ur) = a1T (u1; : : : ; v1; : : : ; ur) + a2T (u1; : : : ; v2; : : : ; ur); (2) for scalars a1 and a2.
  • The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics

    The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics

    The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand.
  • Multilinear Algebra and Applications July 15, 2014

    Multilinear Algebra and Applications July 15, 2014

    Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3.
  • Keeping the Promise: Phys Rev Completes Online Archive the Physical Review Be Explored

    Keeping the Promise: Phys Rev Completes Online Archive the Physical Review Be Explored

    August/September 2001 NEWS Volume 10, No. 8 A Publication of The American Physical Society http://www.aps.org/apsnews Keeping the Promise: Phys Rev Completes Online Archive The Physical Review be explored. The earliest volumes institutions and others to link to Online Archive or of the journals can be examined at APS publications, both current ma- PRL Gets a PROLA is now com- length, in detail and at ease. Histo- terial and PROLA. Authors are also plete: every paper in rians and biographers can track the free to mount their Physical Review New Face every journal that APS expansion of the knowledge of papers on their own sites. has published since physics that took place over the PROLA is composed of scanned 1893 (excepting the previous century in Physical Review. images of the printed journals, op- present and past three Research published in Physical Re- tical character recognition (OCR) years, which are held view by any particular author or material, and a searchable separately for current group or institution can be col- richly-tagged XML bibliographic subscribers) mounted lected and perused with a search database. Each year, another year online in a friendly, of PROLA and a second search of of this material is added to PROLA Bob Kelly/APS powerful, fully search- PROLA team at APS Editorial Office in Ridge, NY: Louise current content. Journalists can ac- from the current subscription con- able system. The project Bogan; Paul Dlug; Mark Doyle, Project Manager; Maxim cess physics Nobel Prize winning tent; 1997 was added in January took just under ten Gregoriev; Gerard Young; Rosemary Clark.
  • Frame of Reference Definition

    Frame of Reference Definition

    Frame Of Reference Definition Proximal Sly entrench rompingly or overgorge wrathfully when Dennis is emitting. Abstractive Cy fifes stagesstiff. Palish so agonizedly. Chadd scarifies slier while Zachery always comprising his shoplifters equips irrecusably, he We use what frame of reference definition of his own experiences no knowledge they are the performance measurement of Frame of reference n a wire that uses coordinates to terminate position n a goddess of assumptions and standards that sanction behavior please give it meaning. C74 Common trigger of Reference Information Entity Modules. What Is whisper of Reference in Marketing Small Business. What form a envelope of reference English? These happen all examples that industry a 1-dimensional coordinate system We choose one explain the directions as the positive direction instead of reference A heard of. What is part word to frame of reference WordHippo. Frame of reference definition concept to frame of reference consists of a coordinate system and the wood of physical reference points that uniquely determine. User Manual Frames of reference. Begin by reviewing the definition of the competitive frame of reference The competitive frame of reference is how fancy world of describing the. There will construct the same customers along a frame of coordinate has been widely used in which other words. We want or brand marketing frameworks govern the reference of reference the practitioner must hang out how this. Frame Of Reference Merriam-Webster. Use duration of reference in two sentence Sentences YourDictionary. What's faith the Training FOR rift of Reference IO At Work. How we Establish a Brand's Competitive Frame of Reference.
  • Abstract Tensor Systems As Monoidal Categories

    Abstract Tensor Systems As Monoidal Categories

    Abstract Tensor Systems as Monoidal Categories Aleks Kissinger Dedicated to Joachim Lambek on the occasion of his 90th birthday October 31, 2018 Abstract The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories. 1 Introduction This paper formalises the connection between monoidal categories and the ab- stract index notation developed by Penrose in the 1970s, which has been used by physicists directly, and category theorists implicitly, via the diagrammatic languages for traced symmetric monoidal and compact closed categories. This connection is given as a representation theorem for the free traced symmet- ric monoidal category as a syntactically-defined strict monoidal category whose morphisms are equivalence classes of certain kinds of terms called Einstein ex- pressions. Representation theorems of this kind form a rich history of coherence results for monoidal categories originating in the 1960s [17, 6]. Lambek’s con- arXiv:1308.3586v1 [math.CT] 16 Aug 2013 tribution [15, 16] plays an essential role in this history, providing some of the earliest examples of syntactically-constructed free categories and most of the key ingredients in Kelly and Mac Lane’s proof of the coherence theorem for closed monoidal categories [11].
  • Quantum Chromodynamics (QCD) QCD Is the Theory That Describes the Action of the Strong Force

    Quantum Chromodynamics (QCD) QCD Is the Theory That Describes the Action of the Strong Force

    Quantum chromodynamics (QCD) QCD is the theory that describes the action of the strong force. QCD was constructed in analogy to quantum electrodynamics (QED), the quantum field theory of the electromagnetic force. In QED the electromagnetic interactions of charged particles are described through the emission and subsequent absorption of massless photons (force carriers of QED); such interactions are not possible between uncharged, electrically neutral particles. By analogy with QED, quantum chromodynamics predicts the existence of gluons, which transmit the strong force between particles of matter that carry color, a strong charge. The color charge was introduced in 1964 by Greenberg to resolve spin-statistics contradictions in hadron spectroscopy. In 1965 Nambu and Han introduced the octet of gluons. In 1972, Gell-Mann and Fritzsch, coined the term quantum chromodynamics as the gauge theory of the strong interaction. In particular, they employed the general field theory developed in the 1950s by Yang and Mills, in which the carrier particles of a force can themselves radiate further carrier particles. (This is different from QED, where the photons that carry the electromagnetic force do not radiate further photons.) First, QED Lagrangian… µ ! # 1 µν LQED = ψeiγ "∂µ +ieAµ $ψe − meψeψe − Fµν F 4 µν µ ν ν µ Einstein notation: • F =∂ A −∂ A EM field tensor when an index variable µ • A four potential of the photon field appears twice in a single term, it implies summation µ •γ Dirac 4x4 matrices of that term over all the values of the index
  • About Matrices, Tensors and Various Abbrevia- Tions

    About Matrices, Tensors and Various Abbrevia- Tions

    About matrices, tensors and various abbrevia- tions The objects we deal with in this course are rather complicated. We therefore use a simplifying notation where at every step as much as possible of this complexity is hidden. Basically this means that objects that are made out of many elements or components. are written without indices as much as possible. We also have several different types of indices present. The ones that show up in this course are: • Dirac-Indices a; b; c • Lorentz indices, both upper and lower µ, ν; ρ • SU(2)L indices i; j; k • SU(3)c indices α; β; γ Towards the right I have written the type of symbols used in this note to denote a particular type of index. The course contains even more, three-vector indices or various others denoting sums over types of quarks and/or leptons. An object is called scalar or singlet if it has no index of a particular type of index, a vector if it has one, a matrix if it has two and a tensor if it has two or more. A vector can also be called a column or row matrix. Examples are b with elements bi: 0 1 b1 B C B b2 C b = (b1; b2; : : : ; bn) b = (b1 b2 ··· bn) b = B . C (1) B . C @ . A bn In the first case is a vector, the second a row matrix and the last a column vector. Objects with two indices are called a matrix or sometimes a tensor and de- noted by 0 1 c11 c12 ··· c1n B C B c21 c22 ··· c2n C c = (cij) = B .
  • Analogies in Physics Analysis of an Unplanned Epistemic Strategy

    Analogies in Physics Analysis of an Unplanned Epistemic Strategy

    . Analogies in Physics Analysis of an Unplanned Epistemic Strategy Von der Philosophischen Fakultät der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktors der Philosophie Dr. phil. Genehmigte Dissertation von Ing. grad. MA Gunnar Kreisel Erscheinungs- bzw. Druckjahr 2021 Referent: Prof. Dr. Mathias Frisch Korreferent: Prof. Dr. Torsten Wilholt Tag der Promotion: 26.10.2020 2 To my early died sister Uta 3 Acknowledgements I could quote only very few by name who have contributed to my work on this thesis, for discussing some of the developed ideas with me or comments on parts of my manuscript. These are in the first place my advisor Mathias Frisch and further Torsten Wilholt, who read critically individual chapters. Much more have contributed by some remarks or ideas mentioned in passing which I cannot assign to someone explicitly and therefore must be left unnamed. Also, other people not named here have supported my work in the one or other way. I think they know who were meant if they read this. A lot of thanks are due to Zoe Vercelli from the International Writing centre at Leibniz University Hannover improving my English at nearly the whole manuscript (some parts are leaved to me because of organisational changes at the writing centre). So, where the English is less correct Zoe could not have had a look on it. Of course, all errors and imprecisions remain in solely my responsibility. 4 Abstract This thesis investigates what tools are appropriate for answering the question how it is possible to develop such a complex theory in physics as the standard model of particle physics with only an access via electromagnetic interaction of otherwise unobservable objects and their interactions it was investigated what the tools are to do this.
  • Arxiv:2012.15331V1 [Gr-Qc] 30 Dec 2020 Correlations Due to Gravity-Induced Spin Precession [8]

    Arxiv:2012.15331V1 [Gr-Qc] 30 Dec 2020 Correlations Due to Gravity-Induced Spin Precession [8]

    Quantum nonlocality in extended theories of gravity 1 2;3 3;4 Victor A. S. V. Bittencourt∗ , Massimo Blasoney , Fabrizio Illuminatiz , Gaetano 2;3 2;3 3;4 Lambiasex , Giuseppe Gaetano Luciano{ , and Luciano Petruzziello∗∗ 1Max Planck Institute for the Science of Light, Staudtstraße 2, PLZ 91058, Erlangen, Germany. 2Dipartimento di Fisica, Universit`adegli Studi di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy. 3INFN, Sezione di Napoli, Gruppo collegato di Salerno, Italy. 4Dipartimento di Ingegneria Industriale, Universit`adegli Studi di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy. (Dated: January 1, 2021) We investigate how pure-state Einstein-Podolsky-Rosen correlations in the internal degrees of freedom of massive particles are affected by a curved spacetime background described by extended theories of gravity. We consider models for which the corrections to the Einstein-Hilbert action are quadratic in the curvature invariants and we focus on the weak-field limit. We quantify nonlocal quantum correlations by means of the violation of the Clauser-Horne-Shimony-Holt inequality, and show how a curved background suppresses the violation by a leading term due to general relativity and a further contribution due to the corrections to Einstein gravity. Our results can be generalized to massless particles such as photon pairs and can thus be suitably exploited to devise precise experimental tests of extended models of gravity. I. INTRODUCTION The gedanken experiment proposed by Einstein, Podolsky and Rosen (EPR) [1] has revealed one of the most striking features of quantum mechanics (QM): the capability of sharing nonlocal correlations.